Final answer:
The greatest possible area of a pen given a fixed perimeter of fencing would be achieved with a square-shaped pen, due to the mathematical property that a square has the largest area of any rectangle for a given perimeter. A) Square-shaped pen is correct answer.
Step-by-step explanation:
The student's question relates to optimizing the area of a geometric figure given a fixed perimeter, specifically in the context of fencing. From the options provided, the square-shaped pen will give the greatest possible number of square feet in the area.
This is based on the mathematical property that for a given perimeter, a square has the largest area of any rectangle. Similarly, among all shapes with a given perimeter, a circle has the greatest area, but since a circular pen is not one of the options, the square pen is the best choice from the given list.
To illustrate this further, for a square pen with side length a, each side would measure the total length of available fencing divided by 4, and the area would be a times a (or a²).
In comparison, a rectangular pen with different length and width would have lesser area, and a triangular pen, even if it is equilateral, would have even less area than the rectangle and square.